Measurement of the W Boson Helicity in Top Quark Decays

by the D0 Collaboration

December 11, 2007

One of the goals of particle physics is
to determine what fundamental particles exist in nature, and the rules
for how they interact with one another. Some, like the electron,
have been known for over a century, but we've continued to find new
ones since then. One of the most recent additions was the top
quark, discovered at Fermilab in 1995 and still the heaviest known
fundamental particle. The
current theory of particle physics, the "Standard Model",
had predicted that this quark must exist, but even so its large mass
came as a surprise, and raises the possibility that the top quark may
show the effects of new physics at energies that we've not yet
directly explored.

In addition to predicting the top quark's existence, the Standard Model
specifies many of the properties of the top quark. In particular,
we know the "rules" a top quark follows when it decays -- that
is, when it transforms into lighter particles. We expect the top
quark to decay to two particles, a W boson and a b quark. What's
more, we can predict detailed features of these particles, even down to
something as seemingly esoteric as the helicity of the W boson.

Helicity is the relationship between the direction in which the particle
is moving and the "spin" of the particle. Spin can be considered
as a directional arrow that's a feature of the particle itself, not of
the particle's motion. For a given type of particle, the length
of this arrow is fixed (we say, for example, that top quarks have "spin
one-half" and W boson have "spin one"), but the direction of the arrow
can vary from one particle to another. Helicity is the relative
direction between the spin arrow and the arrow representing the
particle's motion. For W bosons, there are three possibilities
(spin arrows are red and purple, arrows representing motion are blue
and green):

As the above diagram suggests, we expect that 70% of the W bosons will
have zero helicity, 30% will have negative helicity, and almost
none will have positive helicity. If we were to measure anything
else (for example, if we find a large number of W boson with positive
helicity), we would know that the Standard Model is incorrect -- and that
would be exciting. Our hope is always to expand our knowledge,
and finding an unexpected result is often a good starting point.

Our measurement uses a sample of data collected by the D0 experiment at
Fermi National Accelerator Laboratory from 2002 to 2006.
To make the measurement, we need to do two things: first, pick
out the small number of top quarks in our data from the vast
swarm of other particles, and second, distinguish the various W boson
helicities from one another. For the first step, we're helped by
the fact that top quarks look quite different from most of the other
stuff produced in the experiment. After (years of) improving our
selection techniques we've reached a point where we can pick out top
quarks pretty well.

For the second step, we use a variable known as &theta*, which is illustrated from the vantage point of an observer travelling alongside the W boson:

We know what the relative probability of seeing various values of &theta* for different W boson helicities (&theta*
tends to be small for positive helicity W bosons, large for negative
helicity W bosons, and around 90 degrees for zero helicity W bosons). For technical reasons, the measurement is
easier if we take the cosine of &theta*. All we need to do is compare the values of cos&theta*
in our data sample to the expected distributions. The results,
for a set of data corresponding to one of the signatures the top quark can leave in our detector, are shown below:

In this plot, our data is shown by the points with error bars. The red line represents what we'd see if all the W bosons
had negative helicity, the green line shows the case where they all have
zero helicity, and the blue line shows the case where they all
have positive helicity. The shaded area is the estimated background (meaning
events that don't really have top quarks in them but still sneak into
our sample). Numerically, we
can say that the fractions of W bosons with positive and zero
helicities is:

Fraction with zero helicity: f0 = 0.425 +/- 0.195

Fraction with positive helicity: f+ = 0.119 +/- 0.104

Remember that we expected f0 to be 0.7, and f+
to be very close to zero. So we don't find exactly what we
expected. To see whether we can claim that this means there's
physics beyond the Standard Model, we need to assess the probability of
our result happening just by random chance if the Standard Model is correct. Graphically, the result looks like:

The star shows the Standard Model value, and our measurement is the dot
(the triangle represents the limit of all "reasonable" measurements in
which none of the helicity fractions are negative). The bigger of
the two ellipses is the 95% confidence interval, meaning that if we were to repeat the experiment many times, we'd find an answer outside that ellipse 5% of the time. Since the
Standard Model value is well within this ellipse, our result certainly
cannot be used to claim that the Standard Model is incomplete. In
other words, with the current sample size, it's not
really unlikely that the discrepancy comes about just due to chance --
but we will look again when more data is available. The top quark
may yet hold surprises for us!